AP® Chemistry review sheet from Aim for Five (aimforfive.com/chem/units/3/3-13)
Unit 3 · Topic 3.13
3.13 Beer-Lambert Law
The Beer-Lambert law, A = εbc, says a solution's absorbance depends on how strongly the substance absorbs that wavelength, the path length and the concentration. With wavelength and path length held constant, absorbance is directly proportional to concentration, so a calibration curve lets you find an unknown concentration.
Key terms
- absorbance
- Beer-Lambert law
- molar absorptivity
- path length
- spectrophotometer
- calibration curve
The equation
A = εbc. A is absorbance, a measure of how much light the sample absorbs; it has no units. ε (epsilon) is the molar absorptivity, which describes how strongly a particular substance absorbs light of a particular wavelength, often in M⁻¹cm⁻¹. b is the path length, the distance the light travels through the solution, usually 1.00 cm (the width of the cuvette). c is concentration in mol/L.
Path length and concentration both control how many absorbing particles are in the light's path. Double either one, and the light meets twice as many particles, so the absorbance doubles.
Molar absorptivity is a property of the substance at a given wavelength. A substance with a larger ε gives a bigger absorbance at the same concentration, which is why strongly colored dyes can be measured at very low concentrations.
How a spectrophotometer is used
- Choose the wavelength where the substance absorbs most strongly (λmax). That gives the largest absorbance for a given concentration, so the measurement is most sensitive.
- Zero the instrument with a 'blank': a cuvette containing just the solvent.
- Measure several standards of known concentration.
- Plot absorbance against concentration. This calibration curve (often called a Beer's law plot) should be a straight line through the origin. Its slope equals εb.
- Measure the unknown and read its concentration from the line, or divide its absorbance by the slope.
- Only trust concentrations inside the range of your standards. Very concentrated solutions can fall off the straight line, so dilute the unknown if its absorbance is higher than your most concentrated standard.
Why the wavelength matters
ε changes with wavelength. A substance that absorbs strongly at 520 nm might barely absorb at 420 nm. As long as wavelength and path length stay the same for every measurement, absorbance is proportional to concentration alone.
Colored solutions absorb the colors they don't show. A solution that looks red absorbs mostly blue-green light, so you'd set the spectrophotometer to a blue-green wavelength.
Sources of error
Exam questions often ask how a lab mistake affects the calculated concentration. Work out whether the measured absorbance goes up or down, then follow it through c = A ÷ (εb).
| Mistake | Effect on measured A | Effect on calculated c |
|---|---|---|
| Fingerprints or scratches on the cuvette | higher | too high |
| Cuvette wet with water, diluting the sample | lower | too low |
| Instrument not zeroed with the solvent blank | usually higher | usually too high |
| Wavelength far from λmax (same for standards and unknown) | lower, so less sensitive | less precise, but not shifted in one direction |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Using A = εbc directly
A solution in a 1.00 cm cuvette has an absorbance of 0.452 at a wavelength where ε = 5.60 × 10³ M⁻¹cm⁻¹. What is the concentration?
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- Step 1: Rearrange: c = A ÷ (εb).
- Step 2: c = 0.452 ÷ [(5.60 × 10³ M⁻¹cm⁻¹)(1.00 cm)] = 8.07 × 10⁻⁵ M.
Answer: 8.07 × 10⁻⁵ M
- Example 2Calculator allowed
Using a calibration curve
Standards of a dye at 0, 2.0 × 10⁻⁵, 4.0 × 10⁻⁵, 6.0 × 10⁻⁵ and 8.0 × 10⁻⁵ M give absorbances of 0.000, 0.150, 0.300, 0.450 and 0.600 in a 1.00 cm cuvette. An unknown solution of the dye has A = 0.390. Find its concentration and the molar absorptivity.
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- Step 1: The points form a straight line through the origin. Slope = 0.150 ÷ (2.0 × 10⁻⁵ M) = 7.5 × 10³ M⁻¹.
- Step 2: Slope = εb, so ε = 7.5 × 10³ M⁻¹ ÷ 1.00 cm = 7.5 × 10³ M⁻¹cm⁻¹.
- Step 3: Unknown: c = A ÷ slope = 0.390 ÷ (7.5 × 10³ M⁻¹) = 5.2 × 10⁻⁵ M.
- Step 4: Check: 0.390 is between 0.300 and 0.450, so c should be between 4.0 × 10⁻⁵ and 6.0 × 10⁻⁵ M. It is.
Answer: c = 5.2 × 10⁻⁵ M; ε = 7.5 × 10³ M⁻¹cm⁻¹
- Example 3
Error analysis (classic trap)
A student rinses a cuvette with distilled water and doesn't dry it before adding the unknown solution. How does this affect the calculated concentration of the unknown?
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- Step 1: Leftover water dilutes the sample in the cuvette, so fewer absorbing particles are in the light path.
- Step 2: The measured absorbance is lower than it should be.
- Step 3: Since c = A ÷ (εb), a lower A gives a lower calculated concentration.
- Step 4: The trap is reasoning that 'more liquid means more absorbance'. What matters is concentration, which went down.
Answer: The calculated concentration is too low, because dilution lowers the measured absorbance.
Common mistakes
- Giving absorbance units. It's unitless.
- Choosing a wavelength where the substance absorbs weakly. Use λmax.
- Comparing absorbances measured at different wavelengths or path lengths as if only concentration changed.
- Getting the direction of an error backwards. Decide first what happens to A.
On the exam
- Beer's law is a favorite lab-based free-response topic. Expect to read a calibration curve, calculate a concentration and explain how a procedural error changes the result.
- When asked why a particular wavelength is chosen, say it's where the species absorbs most strongly (largest ε), giving the greatest sensitivity.
Connected topics
Videos
Check yourself
4 questions on 3.13 Beer-Lambert Law. Pick an answer to see if you got it, and why.
| Concentration of dye (M) | Absorbance |
|---|---|
| 0.020 | 0.110 |
| 0.040 | 0.220 |
| 0.060 | 0.330 |
| 0.080 | 0.440 |
| Unknown | 0.275 |
Hypothetical data. A student measures the absorbance of four standard solutions of a dye and one solution of unknown concentration, all at the dye's wavelength of maximum absorbance, using a cuvette with a 1.00 cm path length.
What is the concentration of the dye in the unknown solution?
What is the molar absorptivity, ε, of the dye at this wavelength?
Before measuring the unknown, the student forgets to wipe off fingerprints that absorb some light on the outside of the cuvette. The standards were measured in a clean cuvette. How will this error affect the calculated concentration of the unknown?
A 10.0 mL sample of a colored solution is diluted with water to 50.0 mL. The diluted solution has an absorbance of 0.300 in a 1.00 cm cuvette. The molar absorptivity of the solute at this wavelength is 1.50 × 10³ M⁻¹ cm⁻¹. What was the concentration of the original sample?
0 of 4 answered